IGCSE Mathematics: Probability (Cambridge 0580)

This guide covers Topic 8 Probability, for Cambridge IGCSE Mathematics 0580, 2025–2027 series. The Core subtopics (C8.1–C8.3) are examined at all entry levels; the Extended-only content (E8.1 additions, E8.3 additions, and E8.4, which has no Core equivalent) is required only for the Extended tier. Extended candidates are eligible for grades A* to E and Core candidates for grades C to G, so grades A* to B need the Extended tier (the syllabus advises Extended for anyone aiming at grades A* to C).

Where this fits in 0580

Probability is the eighth of nine topics in 0580, sitting just before Statistics, which it complements rather than depends on directly. Combined-event probability (C8.3/E8.3) draws on basic fraction and decimal arithmetic from Topic 1, so a secure grasp of multiplying and adding fractions pays off directly here, particularly in tree-diagram questions where several branch probabilities must be multiplied and then summed.

Syllabus coverage

CAMBRIDGE IGCSE MATHEMATICS 0580 — TOPIC 8 PROBABILITY

• C8.1 Introduction to probability — understanding and using the probability scale from 0 to 1; calculating the probability of a single event; understanding that the probability of an event not occurring equals 1 minus the probability of it occurring; probability notation is not required at Core, and probabilities should be given as a fraction, decimal or percentage; problems may use information from tables, graphs or Venn diagrams limited to two sets
• C8.2 Relative and expected frequencies — understanding relative frequency as an estimate of probability; calculating expected frequencies; understanding what is meant by fair, bias and random
• C8.3 Probability of combined events — calculating the probability of combined events using sample space diagrams, Venn diagrams and tree diagrams where appropriate; at Core, combined events will only be with replacement, Venn diagrams are limited to two sets, and tree diagram outcomes are written at the end of branches with probabilities by the side of each branch

Note that C8.4 exists only as an Extended-tier subtopic (E8.4 below) — there is no Core content at that number.

Extended only (in addition to the Core content above)

• E8.1 Introduction to probability (Extended) — as C8.1, plus understanding and using probability notation, where P(A) is the probability of A and P(A′) is the probability of not A; Venn diagrams are not restricted to two sets at Extended
• E8.3 Probability of combined events (Extended) — as C8.3, but combined events could be with or without replacement, and the notation P(A ∩ B) and P(A ∪ B) may be used in the context of Venn diagrams
• E8.4 Conditional probability — calculating conditional probability using Venn diagrams, tree diagrams and tables; knowledge of the notation P(A|B) and formulas relating to conditional probability is not required, so this must be handled through direct reasoning with the diagram or table given, not a memorised formula

How to approach it

The single most consequential Core/Extended distinction in this topic is whether combined events are “with replacement” only (Core) or can also be “without replacement” (Extended): in a with-replacement tree diagram, the probabilities on the second set of branches never change, while in a without-replacement diagram, they depend on what happened on the first branch, since an item removed from a group is no longer available to be selected again. Practise identifying which situation a question describes before drawing any branches, since drawing a with-replacement tree for a without- replacement scenario (or vice versa) produces every subsequent calculation with the wrong probabilities. For tree diagrams generally, remember the two governing rules: multiply probabilities along a single path through the branches, and add the probabilities of separate paths that lead to the same outcome. For conditional probability at Extended tier, since no formula or notation is required, the skill being tested is reading the right subset of a Venn diagram or table directly — identifying the reduced sample space implied by the given condition — rather than applying an algebraic rule.

Worked example: without-replacement tree diagram (Extended)

A bag contains 5 red and 3 blue counters. Two counters are drawn without replacement. Find the probability that both are red.

P(first red)  = 5/8
P(second red | first red) = 4/7   (one red counter has been
                                     removed, leaving 7 counters
                                     total, 4 of them red)

P(both red) = 5/8 x 4/7 = 20/56 = 5/14

The key difference from a with-replacement version of the same question is that the second fraction’s denominator drops from 8 to 7 (one counter fewer overall) and its numerator drops from 5 to 4 (one fewer red counter), since the first counter drawn is not returned to the bag.

Common mistakes

Treating a without-replacement scenario as if probabilities stay constant across both draws, copying the first branch’s probabilities onto the second set of branches unchanged. Adding branch probabilities along a single path instead of multiplying them, or multiplying probabilities of separate paths that should be added. Misreading a Venn diagram’s regions when a question asks for a conditional probability, including elements outside the given condition’s subset. Forgetting that C8.1’s probability scale runs from 0 (impossible) to 1 (certain), and giving an answer outside this range as a result of an arithmetic error. Confusing relative frequency (an experimental estimate from repeated trials) with theoretical probability (a calculated value).

Quick revision checklist

• Practise both with-replacement and without-replacement tree diagrams (Extended), correctly adjusting the second set of branch probabilities in the without-replacement case.
• Know when to multiply (along a path) and when to add (across separate paths) in a tree diagram.
• Practise reading probabilities directly from two-set (Core) and more general (Extended) Venn diagrams.
• For Extended conditional probability, practise identifying the correct reduced sample space from a diagram or table without relying on a formula.
• Keep relative frequency (experimental) and probability (theoretical) clearly distinguished in your own explanations.

Official syllabus

Cambridge International, Cambridge IGCSE Mathematics (0580) syllabus for examination in 2025, 2026 and 2027: official syllabus PDF, Subject content, section 8 “Probability”. Verified 2026-09-06.